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Theorem 1stcclb 22606
Description: A property of points in a first-countable topology. (Contributed by Jeff Hankins, 22-Aug-2009.)
Hypothesis
Ref Expression
1stcclb.1 𝑋 = 𝐽
Assertion
Ref Expression
1stcclb ((𝐽 ∈ 1stω ∧ 𝐴𝑋) → ∃𝑥 ∈ 𝒫 𝐽(𝑥 ≼ ω ∧ ∀𝑦𝐽 (𝐴𝑦 → ∃𝑧𝑥 (𝐴𝑧𝑧𝑦))))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐽,𝑦,𝑧   𝑥,𝑋,𝑦,𝑧

Proof of Theorem 1stcclb
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 1stcclb.1 . . . 4 𝑋 = 𝐽
21is1stc2 22604 . . 3 (𝐽 ∈ 1stω ↔ (𝐽 ∈ Top ∧ ∀𝑤𝑋𝑥 ∈ 𝒫 𝐽(𝑥 ≼ ω ∧ ∀𝑦𝐽 (𝑤𝑦 → ∃𝑧𝑥 (𝑤𝑧𝑧𝑦)))))
32simprbi 497 . 2 (𝐽 ∈ 1stω → ∀𝑤𝑋𝑥 ∈ 𝒫 𝐽(𝑥 ≼ ω ∧ ∀𝑦𝐽 (𝑤𝑦 → ∃𝑧𝑥 (𝑤𝑧𝑧𝑦))))
4 eleq1 2828 . . . . . . 7 (𝑤 = 𝐴 → (𝑤𝑦𝐴𝑦))
5 eleq1 2828 . . . . . . . . 9 (𝑤 = 𝐴 → (𝑤𝑧𝐴𝑧))
65anbi1d 630 . . . . . . . 8 (𝑤 = 𝐴 → ((𝑤𝑧𝑧𝑦) ↔ (𝐴𝑧𝑧𝑦)))
76rexbidv 3228 . . . . . . 7 (𝑤 = 𝐴 → (∃𝑧𝑥 (𝑤𝑧𝑧𝑦) ↔ ∃𝑧𝑥 (𝐴𝑧𝑧𝑦)))
84, 7imbi12d 345 . . . . . 6 (𝑤 = 𝐴 → ((𝑤𝑦 → ∃𝑧𝑥 (𝑤𝑧𝑧𝑦)) ↔ (𝐴𝑦 → ∃𝑧𝑥 (𝐴𝑧𝑧𝑦))))
98ralbidv 3123 . . . . 5 (𝑤 = 𝐴 → (∀𝑦𝐽 (𝑤𝑦 → ∃𝑧𝑥 (𝑤𝑧𝑧𝑦)) ↔ ∀𝑦𝐽 (𝐴𝑦 → ∃𝑧𝑥 (𝐴𝑧𝑧𝑦))))
109anbi2d 629 . . . 4 (𝑤 = 𝐴 → ((𝑥 ≼ ω ∧ ∀𝑦𝐽 (𝑤𝑦 → ∃𝑧𝑥 (𝑤𝑧𝑧𝑦))) ↔ (𝑥 ≼ ω ∧ ∀𝑦𝐽 (𝐴𝑦 → ∃𝑧𝑥 (𝐴𝑧𝑧𝑦)))))
1110rexbidv 3228 . . 3 (𝑤 = 𝐴 → (∃𝑥 ∈ 𝒫 𝐽(𝑥 ≼ ω ∧ ∀𝑦𝐽 (𝑤𝑦 → ∃𝑧𝑥 (𝑤𝑧𝑧𝑦))) ↔ ∃𝑥 ∈ 𝒫 𝐽(𝑥 ≼ ω ∧ ∀𝑦𝐽 (𝐴𝑦 → ∃𝑧𝑥 (𝐴𝑧𝑧𝑦)))))
1211rspcv 3556 . 2 (𝐴𝑋 → (∀𝑤𝑋𝑥 ∈ 𝒫 𝐽(𝑥 ≼ ω ∧ ∀𝑦𝐽 (𝑤𝑦 → ∃𝑧𝑥 (𝑤𝑧𝑧𝑦))) → ∃𝑥 ∈ 𝒫 𝐽(𝑥 ≼ ω ∧ ∀𝑦𝐽 (𝐴𝑦 → ∃𝑧𝑥 (𝐴𝑧𝑧𝑦)))))
133, 12mpan9 507 1 ((𝐽 ∈ 1stω ∧ 𝐴𝑋) → ∃𝑥 ∈ 𝒫 𝐽(𝑥 ≼ ω ∧ ∀𝑦𝐽 (𝐴𝑦 → ∃𝑧𝑥 (𝐴𝑧𝑧𝑦))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1542  wcel 2110  wral 3066  wrex 3067  wss 3892  𝒫 cpw 4539   cuni 4845   class class class wbr 5079  ωcom 7707  cdom 8723  Topctop 22053  1stωc1stc 22599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1545  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-in 3899  df-ss 3909  df-pw 4541  df-uni 4846  df-1stc 22601
This theorem is referenced by:  1stcfb  22607  1stcrest  22615  lly1stc  22658  tx1stc  22812
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